Transcription of 1 Eigenvalues and Eigenvectors - Calvin University
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1 eigenvalues and eigenvectors The product Ax of a matrix A Mn n (R) and an n-vector x is itself an n-vector. Of particular interest in many settings (of which differential equations is one) is the following question: For a given matrix A, what are the vectors x for which the product Ax is a scalar multiple of x? That is, what vectors x satisfy the equation Ax = x for some scalar ? It should immediately be clear that, no matter what A and are, the vector x = 0 (that is, the vector whose elements are all zero) satisfies this equation. With such a trivial answer, we might ask the question again in another way: For a given matrix A, what are the nonzero vectors x that satisfy the equation Ax = x for some scalar ?
1 = (1,0,3) and u 2 = (1,1,3). It is a fact that all other eigenvectors associated with λ 2 = −2 are in the span of these two; that is, all others can be written as linear combinations c 1u 1 +c 2u 2 using an appropriate choices of the constants c 1 and c 2. Example: Find the eigenvalues and associated eigenvectors of the matrix A = −1 2 0 ...
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